Let $K$ be a field and consider the quotient of the ring of polynomials in $X$ and $Y$ over $K$ by the ideal generated by $XY$:
$$R:=K[X,Y]/(XY)$$
I have to find the maximal ideals of such $R$. I know the correspondence between (maximal) ideals of $K[X,Y]$, that are all finitely generated, which contain $(XY)$ and (maximal) ideals of $R$ as it is a quotient ring however I'm not sure on how to approach this problem.
If K is algebraically closed the maximal ideals correspond to points on the axes, So they are $$\{(X, Y-a)~|~ a \in K\} \cup \{(X-a, Y)~|~ a \in K\}$$
When K not algebraically closed the answer is messy